Turning a set of scenarios into a return and a risk
The future is rarely a single number. A stock might return 30% in a boom, 8% in a normal year, and −20% in a recession, each with some chance of occurring. The expected return collapses that uncertainty into one figure by weighting every possible return by how likely it is — the more probable a scenario, the more it pulls the average toward its return. But the average alone hides how rough the ride might be, so we also measure thedispersion of outcomes around it. Squaring each deviation from the mean and probability-weighting those gives the variance; its square root, the standard deviation, expresses risk in the same percentage units as the returns themselves.
The expected-return formulas
E(R) = Σ pᵢ · Rᵢ
Var(R) = Σ pᵢ · ( Rᵢ − E(R) )²
σ(R) = √Var(R)
where pᵢ is the probability of scenario i (the probabilities should sum to 1, and this calculator normalises them if they don’t), and Rᵢ is the return in scenario i.E(R) is the expected return, Var(R) the variance, andσ(R) the standard deviation that serves as the risk measure.
What makes this calculator different
- Fully editable scenarios. Add, remove, and rename as many scenarios as your view of the world needs — bull, base, bear, or a dozen finer cases — each with its own probability and return.
- It auto-normalises probabilities. Enter rough odds or relative weights and the calculator rescales them to sum to 1 before computing anything, so the result is always a properly weighted average.
- Both halves of the answer. It reports the expected return and the standard deviation side by side, because reward without its risk tells you only half the story.
- Per-scenario contributions, shown. Each scenario’s weighted contribution to the expected return and its share of the variance are displayed, so you can see exactly which outcomes drive the number rather than trusting a black box.
- Built to connect to theory. The same expected return and standard deviation feed straight into mean-variance portfolio choice and the CAPM calculator, where expected return is tied to systematic risk.
Frequently asked questions
What is expected return and what is the formula?+
Expected return is the probability-weighted average of the returns an investment could deliver across a set of possible scenarios — your best single estimate of the outcome when the future is uncertain. The formula is E(R) = Σ pᵢ·Rᵢ, where pᵢ is the probability of scenario i and Rᵢ is the return in that scenario. You multiply each scenario’s return by its probability and add the products together. The result is not a return you will necessarily earn in any single year; it is the long-run average you would expect if the same probabilities played out many times.
How is the risk (variance and standard deviation) calculated?+
Risk here means how widely the actual outcomes scatter around the expected return. Variance measures that dispersion as the probability-weighted average of the squared deviations from the mean: Var(R) = Σ pᵢ·(Rᵢ − E(R))². Squaring keeps positive and negative deviations from cancelling and penalises large surprises more heavily. Because variance is in “squared return” units, you take its square root to get the standard deviation σ(R) = √Var(R), which is back in the same percentage units as the returns themselves and is the figure most often quoted as risk.
Do the probabilities have to sum to 1?+
In theory, yes — the scenarios are meant to be mutually exclusive and collectively exhaustive, so their probabilities must add to 1 (100%) for the expected return to be a true weighted average. In practice it is easy to enter weights that do not quite add up. This calculator handles that for you: it normalises whatever probabilities you enter so they sum to 1 before computing E(R) and the variance, so you can think in rough odds or relative weights and still get a correctly weighted answer.
What is the difference between expected return and historical average return?+
They answer different questions. Expected return is forward-looking: you assign probabilities to scenarios you believe could happen and weight the returns accordingly. Historical average return is backward-looking: it is the realised average of returns that actually occurred over some past window. The historical average is often used as an input to form expectations, but it is just one sample of how the past unfolded and need not equal the true expected return. Expected return lets you incorporate views — a recession risk, a new product, a regime change — that the historical record cannot yet reflect.
How does expected return feed into CAPM and portfolio choice?+
The expected-return-and-risk pair is the raw material of modern portfolio theory. Investors are assumed to prefer higher expected return and lower standard deviation, so they combine assets to maximise return for a given level of risk, tracing out the efficient frontier. The Capital Asset Pricing Model goes further and says an asset’s required expected return equals the risk-free rate plus its beta times the market risk premium, linking expected return directly to systematic risk. You can explore that relationship in the CAPM calculator. In short, expected return tells you the reward and standard deviation tells you the risk you are taking to earn it.
Disclaimer: This calculator is foreducation and illustration only. Expected return depends entirely on the scenarios and probabilities you supply, which are subjective forecasts, not facts; real outcomes can fall outside any scenario you listed. Nothing here is investment, tax, or trading advice.